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\title[ch05]{Chapter 07: GRADED AND FILTERED MODULES }
\author[]{SCC}
%\institute[XX大学]{XX大学\quad 数学与统计学院\quad 数学与应用数学专业}
%\date{2025年6月}

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% 封面页
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  \titlepage
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% 目录页
\begin{frame}{Contents}
  \tableofcontents
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% Section 1
\section{GRADED RINGS.}
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\begin{frame}{1.1 DEFINITION. }

Let $R$ be a $K$-algebra. We say that $R$ is {\color{red}graded} if there are $K$-vector subspaces $R_i$, $i \in \mathbb{N}$, such that

\begin{enumerate}
    \item $R = \bigoplus_{i \in \mathbb{N}} R_i$;
    \item $R_i \cdot R_j \subseteq R_{i+j}$.
\end{enumerate}

The $R_i$ are called the {\color{red}homogeneous components} of $R$. The elements of $R_i$ are the {\color{red}homogeneous elements} of degree $i$. 

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\begin{frame}{1.2 PROPOSITION. }

Let $R = \bigoplus_{i \geq 0} R_i$ and $S = \bigoplus_{i \geq 0} S_i$ be graded algebras over $K$.
\begin{enumerate}
    \item The kernel of a graded homomorphism of $K$-algebras $\phi : R \longrightarrow S$ is a graded two-sided ideal of $R$.
    \item If $I$ is a graded two-sided ideal of $R$ then $R/I$ is a graded $K$-algebra.
\end{enumerate}

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% Section 2
\section{FILTERED RINGS.}
%---------------------------------------------------
\begin{frame}{2.1 DEFINITION. }

Let $R$ be a $K$-algebra. A family $\mathcal{F} = \{F_i\}_{i \geq 0}$ of $K$-vector spaces is a {\color{red}filtration} of $R$ if

\begin{enumerate}
    \item $F_0 \subseteq F_1 \subseteq F_2 \subseteq \cdots \subseteq R$,
    \item $R = \bigcup_{i \geq 0} F_i$,
    \item $F_i \cdot F_j \subseteq F_{i+j}$.
\end{enumerate}

If an algebra has a filtration it is called a {\color{red}filtered algebra}. 

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\begin{frame}{2.2 EXAMPLE. }

The first filtration of $A_n$ which we will discuss is the {\color{red}Bernstein filtration}. 
It is the filtration defined using the degree of operators in $A_n$. 
Denote by $B_k$ the set of all operators of $A_n$ of degree $\leq k$. 

The Bernstein filtration has a very special feature: each $B_k$ is a vector space of finite dimension. 
A basis for $B_k$ is determined by the monomials $x^\alpha \partial^\beta$ with $|\alpha| + |\beta| \leq k$. 
In particular, $B_0 = K$ and $\{1,x_1,\ldots,x_n,\partial_1,\ldots,\partial_n\}$ is a basis of $B_1$. 

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\begin{frame}{2.3 EXAMPLE. }

Another important example of a filtration for $A_n$ is the {\color{red}order filtration}, denoted by $\mathcal{C}$. 
As in Ch. 3, §2, denote by $C_k$ the vector space of all operators of order $\leq k$ in $A_n$. 
Note that $C_0 = K[X]$ is an infinite dimensional $K$-vector space. 

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% Section 3
\section{ASSOCIATED GRADED ALGEBRA.}

%---------------------------------------------------
\begin{frame}{3.1 DEFINITION. }

Let $R$ be a $K$-algebra. Suppose that $\mathcal{F} = \{F_i\}_{i \in \mathbb{N}}$ is a filtration of $R$. As a first step in the construction of the graded algebra, we introduce the {\color{red}symbol map} of order $k$, which is the canonical projection of vector spaces

$$
\sigma_k : F_k \longrightarrow F_k/F_{k-1}.
$$

Thus for an operator $d \in F_k$, the {\color{red}symbol} $\sigma_k(d)$ is non-zero if and only if $d \notin F_{k-1}$.

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\begin{frame}{3.2 DEFINITION. }

Consider now the $K$-vector space
$$
gr^{\mathcal{F}} R = \bigoplus_{i \geq 0} (F_i/F_{i-1}).
$$

A homogeneous element of $gr^{\mathcal{F}} R$ is of the form $\sigma_k(a)$ for some $a \in F_k$. 
Let $\sigma_m(b)$, for $b \in F_m$ be another homogeneous element, and define their product by
$$
\sigma_k(a) \sigma_m(b) = \sigma_{m+k}(ab).
$$

A straightforward verification shows that $gr^{\mathcal{F}} R$ with this multiplication is a graded $K$-algebra, with homogeneous components $F_i/F_{i-1}$. 

This is called the {\color{red}graded algebra} of $R$ associated with the filtration $\mathcal{F}$.

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\begin{frame}{3.3 THEOREM. }

The graded algebra $S_n = gr^{\mathcal{B}} A_n$ is isomorphic to the polynomial ring over $K$ in $2n$ variables.

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% Section 4
\section{FILTERED MODULES.}
%---------------------------------------------------
\begin{frame}{4.1 DEFINITION. }

Let $M$ be a left $A_n$-module. A family $\Gamma = \{\Gamma_i\}_{i \geq 0}$ of $K$-vector spaces of $M$ is a {\color{red}filtration of $M$} if it satisfies

\begin{enumerate}
    \item $\Gamma_0 \subseteq \Gamma_1 \subseteq \cdots \subseteq M$,
    \item $\bigcup_{i \geq 0} \Gamma_i = M$,
    \item $B_i \Gamma_j \subseteq \Gamma_{i+j}$.
\end{enumerate}

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% Section 5
\section{INDUCED FILTRATIONS.}
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\begin{frame}{5.1 LEMMA. }

Let $M$ be an $A_n$-module with a filtration $\Gamma$ compatible with $\mathcal{B}$. The sequence of $S_n$-modules
$$
0 \rightarrow gr^{\Gamma'} N \xrightarrow{\phi} gr^\Gamma M \xrightarrow{\pi} gr^{\Gamma''} M/N \rightarrow 0
$$
is exact.

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% Section 6
\section{EXERCISES.}
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\begin{frame}{EXERCISE 1. }

Let $F = K\{x,y\}$ be the free algebra in two generators, and $I$ the two-sided ideal of $F$ generated by the relation $xy - \lambda yx$. Show that the quotient ring $F/I$ is a graded ring.

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\begin{frame}{EXERCISE 2. }

Let $R = \bigoplus_{i \geq 0} R_i$ be a graded ring and $M = \bigoplus_{i \geq 0} M_i$ a graded $R$-module. Let $M(k)_i = M_{i-k}$.

\begin{enumerate}
    \item Show that $M(k) = \bigoplus_{i \geq 0} M(k)_i$ is a graded $R$-module.
    \item Show that if $k > 0$ then the identity map $M \longrightarrow M(k)$ is an isomorphism of $R$-modules, but not a graded isomorphism.
\end{enumerate}

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\begin{frame}{EXERCISE 3. }

In this exercise we define a grading for $A_1$ that is not positive. Denote by $x$ and $\partial$ the generators of $A_1$. Define $G_k = \{d \in A_1 : [x\partial, d] = kd\}$. Let $K[x\partial]$ be the polynomial ring in the operator $x\partial$. Show that

\begin{enumerate}
    \item $G_k = K[x\partial] x^k$ for $k \geq 0$,
    \item $G_k = K[x\partial] \partial^{-k}$ for $k \leq 0$,
    \item $A_1 = \bigoplus_{k \in \mathbb{Z}} G_k$ is a graded ring.
\end{enumerate}


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\begin{frame}{EXERCISE 4. }

Let $d \in A_n$. Define the \textit{principal symbol} of $d$ by $\sigma(d) = \sigma_k(d)$, where $k$ is the degree of $d$ and $\sigma_k$ denotes the symbol map of degree $k$ relative to the Bernstein filtration. Find the principal symbol of the following operators of $A_3$:

\begin{enumerate}
    \item $\partial_1^6 x_2^6 + x_3^5$,
    \item $x_1^6 \partial_2^6 + x_2 x_3^3 \partial_1^2 \partial_3 + x_2 x_4 \partial_3^2 + x_5^6$,
    \item $\partial_2^7 + x_3^7 + x_1^4 x_2^3 + \partial_2 x_3^5 + x_2$.
\end{enumerate}


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\begin{frame}{EXERCISE 5. }

Let $\tau_k$ be the symbol map of order $k$ with respect to the order filtration of $A_n$. Let $\xi_i = \tau_1(\partial_i)$. Show that $gr^\mathcal{F} A_n$ is isomorphic to the polynomial ring $K[X][\xi_1,\ldots,\xi_n]$.


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\begin{frame}{EXERCISE 6. }

Let $R$ be a filtered $K$-algebra with a filtration $\mathcal{F}$. Show that if $gr^\mathcal{F} R$ is a domain, then so is $R$.


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\begin{frame}{EXERCISE 7. }

Let $J$ be a left ideal of $A_n$. Define the \textit{symbol ideal} of $J$ to be the ideal $gr(J) = \sum_{k \geq 0} \sigma_k(J \cap B_k)$ of $S_n$. Let $M = A_n / J$.

\begin{enumerate}
    \item Let $\mathcal{B}'$ be the filtration of $J$ induced by the Bernstein filtration. Show that $gr^{\mathcal{B}'} J \cong gr(J)$.
    \item Show that if $\mathcal{B}''$ is the filtration of $M$ induced by the Bernstein filtration, then $gr^{\mathcal{B}''} M \cong S_n / gr(J)$.
\end{enumerate}


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\begin{frame}{EXERCISE 8. }

The definition of a filtered $A_n$-module for the order filtration is analogous to that given in §4, except that (4) must be replaced by: $M_i$ is a finitely generated $K[X]$-module, for $i \geq 0$. Find a filtration of the $A_n$-module $K[X]$ with respect to the order filtration and calculate its associated graded module.


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\begin{frame}{EXERCISE 9. }

Let $V$ be the $K$-vector subspace of $A_n$ with basis $\{x_1,\ldots,x_n,\partial_1,\ldots,\partial_n\}$. Let $Sp(V)$ be the symplectic group on $V$.

\begin{enumerate}
    \item Show that an element $\sigma \in Sp(V)$ can be extended to an automorphism of $A_n$ that preserves the Bernstein filtration.
    \item Show that this automorphism induces an automorphism of $S_n = gr^{\mathcal{B}} A_n$.
\end{enumerate}


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\begin{frame}{EXERCISE 10. }

Let $(\lambda_1,\ldots,\lambda_{2n}) \in K^{2n}$. Show that the formulae

$$
\sigma(x_i) = x_i - \lambda_i
$$

and

$$
\sigma(\partial_i) = \partial_i - \lambda_{n+i}
$$

define an automorphism of $A_n(K)$ that preserves the Bernstein filtration and induces the identity automorphism on $S_n$.


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\begin{frame}{EXERCISE 11. }

Let $d \in A_n(\mathbb{C})$ be an element of degree 2. Show that there exist an automorphism $\alpha$ of $A_n(\mathbb{C})$ and constants $\lambda_1,\ldots,\lambda_n,r_1,\ldots,r_n$ such that

$$
\alpha(d) = \sum_{i=1}^{n} \lambda_i (x_i^2 + \partial_i^2) + r_i.
$$

Hint: Use Exercises 6.9 and 6.10, and the fact that quadratic forms are diagonalizable.


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